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# Mailing incongruent cubes to a friend

6 replies to this topic

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Posted 16 October 2013 - 05:58 PM

Call a set of cubes incongruent if they all have different side lengths. Prove that it is impossible to exactly fill a rectangular box with incongruent cubes.

Note: The phrase "exactly fill" means that there is no space in the box which is not occupied by a cube, and that the cubes themselves should be packed together to form the shape of the rectangular box that envelops them.

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### #2 jamieg

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Posted 16 October 2013 - 09:47 PM

Trivial case excluded?

(Rectangular box of sides nxnxn, filled with a single cube of sides nxnxn, no two cubes have the same length)

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### #3 bonanova

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Posted 16 October 2013 - 11:09 PM   Best Answer

Spoiler for Proof

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### #4 Anza Power

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Posted 16 October 2013 - 11:11 PM

Spoiler for

Edited by Anza Power, 16 October 2013 - 11:13 PM.

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### #5 bonanova

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Posted 17 October 2013 - 10:36 AM

Spoiler for

It works in 2D. The smallest square just can't be on an edge.
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### #6 phil1882

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Posted 17 October 2013 - 02:52 PM

could you give a pictorial example of it working in 2d bonanova?

i can't find an example.

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### #7 bonanova

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Posted 17 October 2013 - 06:17 PM

could you give a pictorial example of it working in 2d bonanova?
i can't find an example.

Spoiler for For example

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